A Kolmogorov inequality for the sum of independent Bernoulli random variables with unequal means (Q1892956)

Let \(\{X_ i\}\) be a sequence of independent Bernoulli random variables such that \(X_ i\) has parameter \(p_ i\) with \(0 \leq p_ i \leq 1\). Let \(\overline X_ k = (1/k) \sum_ 1^ k X_ i\) and let \(\overline p_ k = (1/k) \sum^ k_ 1 p_ i\). Then for any constant \(\varepsilon \geq 0\) and any number \(n \geq 0\), \(P(\sup_{k \geq n} (\overline X_ k - \overline p_ k) \geq \varepsilon) \leq e^{-2n \varepsilon^ 2}\). This is the extension of the Kolmogorov inequality announced in 1963 without proof.